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Newton second law for the motion of an electron through a ring of charge

H.Tahsiri

Off[General::spell];
Off[General::spell1];
Pagewidth->70;
Clear["Global`*"];
Clear[y,V,a,t]

1) The electric field near y=0 is

Ep = k Q y/R^3

     k Q y
     -----
       3
      R

2) Use Newton second law to write the equation of motion

F = ma;a=dV/dt

3) The force and the field are related by

F=q Ep =-e Ep

4) The angular frequency and the period are related by

w0=2Pi f=2Pi/T= k Q e/(m R^3)=1

5 ) The equation of motion is written as

equ= D[y[t],{t,2}]+ y[t]

     y[t] + y''[t]

6 ) The Solution is

sol=DSolve[{equ==0,y[0]==y0,y'[0]==0},y[t],t][[1,1]]

     y[t] -> y0 Cos[t]

7 ) The displacement of the electron as a function of time is

y[t_]=y0 Cos[t]

     y0 Cos[t]

8 ) The velocity of the electron as a function of time is

V[t_]=D[y[t],t]

     -(y0 Sin[t])

9 ) The acceleration of the electron as a function of time is

a[t_]=D[V[t],t]

     -(y0 Cos[t])

10 ) Plot the displacement, the velocity and the acceleration

ploty=Plot[y[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[1,0,0]];

Plot of Displacement verses Time: y(t)=y0 sin(t)

Plot of Velocity verses Time: v(t)= - y0 sin(t)

plotV=Plot[V[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[0,0,1]];

Plot of Acceleration verses Time: a(t)= - y0 cos(t)

plota=Plot[a[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[0,1,0]];

11 ) Combine the plots

plotall=Show[{ploty,plotV,plota},PlotRange->{-.1,.1}];

y(t)=y0 cos(t)

v(t)= - y0 sin(t)

a(t)= - y0 cos(t)

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